In 2003 Kirk-Srinavasan-Veeramani [1] was introduced the notion of cyclic contraction mapping and proved some fixed point theorems for the operators in the class of cyclic contraction. Since then many interesting fixed point theorems for contractive type conditions have been obtained by various authors. The existence and uniqueness of fixed point and common fixed point theorems of operators has been subject of great interest since Banach [2] proved the Banach contraction principle in 1922. A great number of studies concerning fixed points of contractions on different spaces have been represented among those spaces are the metric spaces, quasi-metric space, Cone metric space, Fuzzy metric space, Menger spaces.

Let (X,d) be a metric space. A mapping is a contraction if there exists a constant such that

holds for any x, y 2 X. If X is complete, then every contraction has a unique fixed point and that point can be obtained as a limit of repeated iteration of the mapping at any point of X. Obviously, every contraction is a continuous function. Generalization of the above principle has been a heavily investigated branch of research. In [3] Boyd and Wong proved that the constant k can be replaced by the use of upper semi-continuous function. In ([4], [5]) generalized Banach contraction conjecture has been established. In [11] Suzuki has proved a generalization of the same principle which characterizes metric completeness. The contraction principle has also been extended to probabilistic metric space [7].

Let (X, d) be a metric space. Suppose A and B are nonempty subsets of X. A mapping: is called a cyclic map if and We give the following basic definitions and some important results, which will be used in the our main result.

Definition 1.1 [8]. A function is called a distance function if the following properties are satisfied :

(a) ,

(b) is continuous and monotonically non-decreasing.

Definition 1.2 [9]. Let (X, d) be metric space, m a positive integer, closed non-empty subsets of X and . An operator is called a cyclic weak -contraction if

(i) is a cyclic representation of Y with respect to T, and

(ii) there exits a continuous, non-decreasing function with

for t > 0 and , such that

, (1.1)

for any , , i = 1, 2, ...,m where .

Notice that the definition above generalizes cyclic -contraction introduced by Pacurar and Rus [10] by suitable choice of . Karapnar [9] proved the following theorem for the maps given in the previous definition.

Theorem 1.3. Let (X, d) be a complete metric space, closed non-empty subsets of X and . Let be a cyclic generalized weak -contractive mapping, where is a nondecreasing and continuous function with for all , and (0) = 0. Then T has a unique fixed point .

We denote by the class of all function such that satisfies the following condition:

Obviously, for every if then by defined we conclude that and hence . Therefore t = 0. For example, every nondecreasing function with for belong to .

2. MAIN RESULTS:

In this paper, we introduced the definition of generalized cyclic − contraction and proved unique fixed point for the contractive condition.

Definition 2.1. Let (X, d) be a metric space m a positive integer closed nonempty subsets of X and . An operator is called a generalized cyclic − contraction if

(i) is a cyclic representation of Y with respect to T, and

(ii) there exists two function where and is continuous such that

, (2.1)

for all any for any , , i = 1, 2, ...,m where , and where

Theorem 2.2. Let closed nonempty subsets of a complete metric space (X, d). Suppose , is continuous and be a generalized cyclic − contraction mapping satisfying (2.1). Then T has a unique fixed point .

Proof. Suppose an arbitrary and construct the sequence such that Now using (2.1), for all , we get

(2.2)

where

(2.3)

If then from (2.2) and (2.3), we obtain

(2.4)

It is clear that

(2.5)

and it follows that

and this is a contradiction. Therefore , for all . Thus the sequence is decreasing sequence of nonnegative number. So there exits for some such that

. (2.6)

Making in (2.4) and using continuity of and , we have

which is a contradiction. Hence

. (2.7)

Next, we show that is Caucny sequence in (X, d). Then there exists for which we can find subsequences and of with such that for every i,

, (2.8)

and

, (2.9)

From (2.8) and (2.9) and the triangular inequality, we have

Taking in the above inequality and using (2.8), we get

. (2.10)

Again from (2.8) and the triangular inequality, we get that

On letting in the above inequalities and using (2.9) and (2.10), we have

(2.11)

From (2.3), we obtain

(2.12)

where

From (2.11) and (2.13), we get

(2.14)

So that from (2.11), (2.12) and (2.14), we calculate that

. (2.15)

Since , then , which is a contradiction.

This shows that is a Caucny sequence and hence is convergent in the complete matric space X. Suppose , as . substituting and in (2.1), we get

where

Since (X, d) is complete and the sequence is Cauchy, there exists x in X such that . We calculate that for i = 1, 2, ..,m since ,

and is closed. So . For all from (2.1) and , we have

(2.16)

where

This inequality shows that . We need to show that .

If then for , there exists such that for all , .

Since is nondecreasing, from (2.16)

(2.17)

for all . Letting in above inequality, we obtain

(2.18)

and this implies that . This is a contradiction. Hence , therefore T has a common fixed point. Uniqueness of the common fixed point follows from 3.1, and this completes the proof.

3. REFERENCES:

[1] Kirk, W.A., Srinavasan, P.S. and Veeramani, P., Fixed points for mapping satisfying cyclical contractive conditions, Fixed point theory, 4 (2003), 79-89.

[2] Banach, S., Surles operation dans les ensembles ab- stracts etleur application aux equations integrals.Fun.Math,Vol. 3, (1922), 133 - 181.

[3] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proceedings of the American Mathematical Society, vol. 20, no. 2, pp. 458464, 1969.

[4] A.D. Arvanitakis, A proof of the generalized Banach contraction conjecture, Proceedings of the American Mathematical Society, vol. 131, no. 12, pp. 36473656, 2003.

[5] J. Merryfield, B. Rothschild, and J. D. Stein Jr., An application of Ramseys theorem to the Banach contraction principle, Proceedings of the American Mathematical Society, vol. 130, no. 4, pp. 927933, 2002.

[6] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 18611869, 2008.

[7] O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

[8] M.S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bulletin of the Australian Mathematical Society, vol. 30, no. 1, pp. 19, 1984.

[9] Karapinar, E., Fixed point theory for cyclic weak _-contraction, Appl. Math. Lett. 24(2011), 822-825.

[10] Pacurar, M. and Rus, I.A., Fixed point theory for cyclic psi-contraction, Nonlinear Anal.(TMA) 72 (2010), 1181-1187.

[11] Suzuki, T: A generalized Banach contraction principle that characterizes metric completeness. Proc Am Math Soc. 136, 1861186 (2008)